Banach-Valued Multilinear Singular Integrals with Modulation Invariance

We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded L p-extension to triples of intermediate operatorname UMD spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of UMD spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the UMD-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the UMD-valued bound for bilinear Calderón-Zygmund operators recently obtained by the same authors.